Algorithms in Bhāskara’s commentary on Āryabhat. īya 491
Th e quotient of that divided by the measure should be this fruit of desire|| 12
In other words, if M (the measure) produces a fruit F M , and D is a desire
for which the fruit F D is sought, the verse may be expressed in modern
algebraic notation as:
F
FD
D M
= M× (1)
Obviously, this expression can also be understood as a statement that the
ratios are equal:
F
D
F
M
DM= (2)
Th e procedure given in the verse provides an order for the diff erent
operations to be carried out. First, the desire is multiplied by the fruit. Next,
the result is divided by the measure. Th is order of operations causes the
procedure to appear as an arbitrary set of operations.^13 Bhāskara provides a
standard expression to defi ne the kind of problem which the Rule of Th ree
solves. When the commentator thinks that a situation involves propor-
tional quantities and thus the Rule of Th ree is (or can be) applied, he brings
this fact to light by using a verbal formulation ( vāco yukti ) of the Rule of
Th ree. Th is verbal formulation is a syntactically rigid question which reads
as follows:
If the measure produces the fruit, then with the desire what is produced? Th e fruit
of desire is produced.
Th is question, when it appears, shows that Bhāskara thinks that the Rule of
Th ree can be applied. I believe that for Bhāskara the Rule of Th ree invokes
proportionality.
2.2 Th e Pythagorean Th eorem
Bhāskara, like other medieval Sanskrit mathematicians, does not use the
concept of angles. In his trigonometry, Bhāskara uses lengths of arcs. As
for right-angled triangles, Bhāskara distinguishes them from ordinary
triangles by giving to each side a specifi c name. Whereas scalene, isosceles
12 trairāśikaphalarāśim. tam athecchārāśinā hatam. kr. tvā|
labdham. pramān. abhajitam. tasmād icchāphalam idam. syāt|| (Shukla 1976 : 115–223).
13 If the division was made fi rst (resulting in the ‘fruit’ of one measure) and then the
multiplication, the computation would have had a step-by-step meaning, but this is not the
order adopted by Ab.